Highschool Exam Question About Cube Factoring

Question

Given;
$ a^3 - 3ab^2 = 10 $ and $ b^3 - 3ba^2 = 5$

What is the value of $ a^2 + b^2 $ ?

Answer 1

Hint: $$(a+bi)^3=(a^3-3a^2b)-i(b^3-3a^2b)=10-5i$$

$$a^2+b^2=|a+bi|^2$$

Answer 2

\begin{align} 10 &= a^3 - 3ab^2 \\ 5 &= b^3 - 3a^2b \\ \hline 100 &= a^6 -6a^4b^2 + 9a^2b^4 \\ 25 &= b^6 - 6a^2b^4 + 9a^4b^2 \\ \hline 125 &= a^6 + 3a^4b^2 + 3a^2b^4 + b^6 \\ 5^3 &= (a^2 + b^2)^3 \\ 5 &= a^2 + b^2 \end{align}